Georgia Institute of Technology College of Sciences

Using the theory of total linear stability for Dynkin quivers and an interplay between the Bruhat order and the noncrossing partition lattice, we define a family of triangulations of the permutohedron indexed by Coxeter elements.  Each triangulation is constructed to give an explicit homotopy between two complexes (the Salvetti complex and the Bessis--Brady--Watt complex) associated to two different presentations of the corresponding braid group (the standard Artin presentation and Bessis's dual presentation).  Our triangulations have several notable combinatorial properties. In addition, they refine similar Bruhat interval polytope decompositions of Knutson, Sanchez, and Sherman-Bennett. This is based on joint work with Melissa Sherman-Bennett and Nathan Williams. 

Let $f,g$ be $C^2$ area-preserving Anosov diffeomorphisms on $\mathbb{T}^2$ which are topologically conjugated by a homeomorphism $h$. It was proved by de la Llave in 1992 that the conjugacy $h$ is automatically $C^{1+}$ if and only if the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all periodic orbits. We prove that if the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all points of some large period $N\in\mathbb{N}$, then $f$ and $g$ are ``approximately smoothly conjugate." That is, there exists a a $C^{1+\alpha}$ diffeomorphism $\overline{h}_N$ that is exponentially close to $h$ in the $C^0$ norm, and such that $f$ and $f_N:=\overline{h}_N^{-1}\circ g\circ \overline{h}_N$ is exponentially close to $f$ in the $C^1$ norm.

Zoom link - 

https://gatech.zoom.us/j/5506889191?pwd=jIjsRmRrKjUWYANogxZ2Jp1SYdaejU.1

Meeting ID: 550 688 9191

Passcode: 604975

Mean-field games (MFG) theory is a mathematical framework for studying large systems of agents who play differential games. In the PDE form, MFG reduces to a Hamilton-Jacobi equation coupled with a continuity or Kolmogorov-Fokker-Planck equation. Theoretical analysis and computational methods for these systems are challenging due to the absence of strong regularizing mechanisms and coupling between two nonlinear PDE.

One approach that proved successful from both theoretical and computational perspectives is the variational approach, which interprets the PDE system as KKT conditions for suitable convex energy. MFG systems that admit such representations are called potential systems and are closely related to the dynamic formulation of the optimal transportation problem due to Benamou-Brenier. Unfortunately, not all MFG systems are potential systems, limiting the scope of their applications.

I will present a new approach to tackle non-potential systems by providing a suitable interpretation of the Benamou-Brenier approach in terms of monotone inclusions. In particular, I will present advances on the discrete level and numerical analysis and discuss prospects for the PDE analysis.